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1 [ DOI:.68/IJCE... ] WaterHydraulic Structure Estimation of maximum scour depth donstream of horizontal and adverse stilling basins using a semi-theoretical approach H. Khalili Shayan, J. Farhoudi, H. Hamidifar Received: June, Revised: July, Accepted: October Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 Abstract Because of the complexity of the physical processes in the vicinity of the hydraulic structures due to the separation of the flo, traditional methods for for prediction of maximum scour depth donstream of hydraulic structures are mostly based on empirical approaches. Hence, only a fe theoretical orks have been reported to study this phenomenon. The present paper describes a ne approach based on the momentum principles to estimate the maximum local scour depth donstream of a submerged sluice gate floing over horizontal or adverse stilling basin. A control volume of the fluid in the euilibrium state of the scour hole as considered and based on momentum principles, some euations are derived to estimate the scour depth at euilibrium state. To verify the proposed euations, large numbers of experiments ere planned and conducted under ide range of characteristic parameters such as, incoming Froude number, sediment size, tailater depth, length and slope of the apron. It as found that the proposed euations fall in a good agreement ith experimental results. It as also observed that, in the case of horizontal apron, a specific tailater depth exists ith hich the local scour depth attains a minimum value. Hoever, in the case of adverse basins hen the tailater depth takes a specific value, the maximum depth of the scour hole reaches to its maximum and then decreases to a constant value as the tailater depth increases. This critical tailater depth as formulated using a semi-theoretical euation. Keyords: Local scour, Momentum principle, Sluice gate, Submerged jet, Adverse Stilling basin.. Introduction Local scour problem donstream of hydraulic structures is a serious problem hich endanger the stability of such structures leading to undermine the foundation. Hence, the geometrical characteristics of scour hole is too significant to be considered by designers of hydraulic structures such as, stilling basins and sluice gates. The presence of large scale turbulence, local eddies encountered ith hydraulic jump and increasing shear stress results in formation of the scour hole donstream of hydraulic structures. Because of the complexity of the physical process, many researchers have employed empirical approaches to study this phenomenon. Farhoudi and Smith [,] studied the scour process donstream of hydraulic jump featuring the characteristic parameters defining the scour hole. Corresponding author: jfarhoudi@ut.ac.ir Postgraduate, Irrigation and Reclamation Engineering Department, University of Tehran, P.O. Box 587-4, Karaj, Iran Professor, Irrigation and Reclamation Engineering Department, University of Tehran, P.O. Box 587-4, Karaj, Iran Faculty Member, Water Engineering Department, Fasa University, P.O. Box , Fasa, Iran They demonstrated that the development of local scour hole donstream of apron in the passage of time shos a certain geometrical similarity and nondimensional scour profile can be presented by a unified euation. Hassan and Narayanan [] studied the local scour donstream of an apron due to a submerged jet issuing from a sluice opening and developed an empirical euation for the time variation of scour depth to reach an asymptotic stage. Chatterjee et al. [4] studied the local scour donstream of an apron due to a submerged jet issuing from a sluice opening and developed an empirical euation for the time variation of scour depth to reach an asymptotic stage. Balachandar et al. [5] investigated the effect of grain size on local channel scour belo a sluice gate. They have found that the tests ith a mixed sand bed resulted in less scour relative to those in hich a uniformly graded sand of similar grain size as used. Dey and Sarkar [6] carried out an experimental investigation on the effects of different parameters on scour depth due to submerged horizontal jets. A particular geometrical similarity of scour profiles at different times have been exhibited and expressed by a combination of to polynomials. Dey and Sarkar [7] investigated the effects of upard seepage on scour hole at the donstream an apron due to submerged jets. They reported that the maximum scour depth increases linearly ith an increase in upard seepage velocity. Oliveto et al. [8] shoed that the main International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5

2 [ DOI:.68/IJCE... ] dimensionless parameters governing the scour process are the tailater densimetric Froude number, relative time, and relative tailater flo depth. Many investigations have been conducted regarding the evolution of local scour in last three decades hich are summarized in Table, here is flo depth at the vena contracta, is the angle of repose, D5 is the median sediment size, Fr g is the approaching Froude number, ss is the relative density of sediments, LB is the length of stilling basin, is the tailater depth, gate opening, Fo Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 number, is g ss D5 the gate uo is the jet velocity at the is the densimetric Froude opening, D9 is the sedimentological diameter, and u is the flo velocity at the donstream section. Despite to the several above mentioned orks, it seems that the effects of some important parameters such as tailater depth, length and slope of the stilling basin are not still ell understood. Therefore, the local scour phenomenon ould be one of the concerns in engineering applications. It should be noted that the mentioned relations have been proposed in certain conditions. For example, Altinbilek and Basmaki [9] and Hoffmans [] proposed some relationships for scour evolution donstream of the submerged sluice gates floing over aterays ithout any bed protections. On the other hand, Dey and Sarkar [6] derived a relationship from a set of experimental data here some relatively short aprons (maximum relative length of apron ( LB ) as eual to 55) ere employed. Hamidifar et al. [] performed some experiments on a stilling basin ith sufficient long apron. Apparently, it ould be concluded that, the suggested euations by Dey and Sarkar [6] and Hamidifar et al. [] may not correctly estimate the geometry of local scour hole here aprons ith different lengths are used. Table Proposed euations to estimate the maximum euilibrium scour depth donstream of submerged sluice gates, by different authors Researchers Proposed relation.5 Altinbilek and Basmaki [9] Chatterjee et al. [4] Hoffmans [] Dey and Sarkar [6] Hamidifar et al. [] It is understood that any changes in channel geometry, such as adverse slope of stilling basin, ould change the characteristics of hydraulic jump and therefore, its conseuences at donstream of such structures. Since the adverse aprons ould increase the energy loss and decrease the length of hydraulic jump, it could be recommended as an economical replacement to classical stilling basins hich detailed by, McCoruodale and Mohamed [], Pagliara and Peruginelli []. On the other hand the hydraulic jump on such aprons accompanies ith some instabilities on hydraulic jump exiting from the basin (Rajaratnam [4], Pagliara and Peruginelli [], Baines and Whitehead [5]). Therefore, the adverse aprons may act dual roles in evolution of scour hole donstream of such stilling basins hich reuires a close study to achieve some recommendations for designers. In this research, some ne regression relations are presented in those a ide range of parameters ere considered. It is clear that, regression euations entirely tan Fr D5 ss U.775 o g 5 D 9 L.59 Fo.94 B u u.7 y. Fo t.6 D5.5.8 depends on experimental data. Using the momentum euation ould facilitate the derivation of some semitheoretical relationships to explain the real phenomenon. The momentum principles ere used to derive some theoretical euations to predict the maximum scour depth at presence of a horizontal and adverse stilling basin. At the end, the semi-theoretical euations are compared ith those of regression relationships hich shos that the semi-theoretical approach is capable to predict the specific depth affecting the characteristics of scour hole.. Material and Methods To evaluate the characteristics of local scour donstream of a horizontal and adverse stilling basin, a perspex-alled flume of 4.9 m length and.4 m idth ith a recirculation flo system as employed here a submerged sluice gate, ith its opening ( ) of.4,, and H. Khalili Shayan, J. Farhoudi, H. Hamidifar

3 Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 [ DOI:.68/IJCE... ].5 cm, as erected at upstream of.8 m perspex apron. The details of experimental layouts are shon in Fig.. Seven bed slopes ( S o tan ) of.56,.956,.764,.5,.6,.9, and ere used here an alluvial chamber ith.5 m depth and.8 m length as constructed across the hole idth of the flume donstream of the stilling basin. A sand trap as placed donstream of the alluvial bed to prevent any incidental fine sand intrusion into the recycling flo. Totally, tests in a ide range of sediment size ( D5 ), relative L length of stilling basin ( B ), bed slope ( So tan ), Fr inflo Froude number ( ) and relative tailater depth ( ) ere performed as is summarized in Table. Fig. Experimental layout; (a) setup, (b) Typical sketch of the scour hole Table Ranges of different parameters in this study repemarap Range of variation D5 (mm) So tan LB Fr.58(D4),.(D),.78(D).56,.956,.764,.5,.6,.9, 47.,59.5, Figure shos the sediment size distribution of the three used sediments throughout the experiments. From Table, it ould be realized that the sediments could be assumed as uniformly graded. A hinged tailgate as used for adjusting the tailater depth. The discharge as measured by a V-notch eir hich as calibrated against an electromagnetic flometer. The tailater depth as measured by means of a moving point gauge ith an accuracy of ±. mm. In order to avoid the undesirable erosion of sediment at the beginning of each experiment, the flume as initially filled ith ater from the donstream section. Once the ater level reached to a desired depth, the experiment as commenced. The profiles of scour holes ere traced by means of digital International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5

4 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 photography techniue. On the basis of the previous studies conducted by Farhoudi and Smith [,], each test as carried out over a period of 4 hours. Although, the euilibrium sour state as not achieved in this time length, but it as sufficient to reach a uasi euilibrium state of scouring in most of the tests. Figure shos the typical time variation of scour holes for ( LB So.56, Fr 8.5, D5.mm, 84. ). Fig. Distributions of particle size Fig. Typical time variation of scour hole profiles for S o.56, Fr 8.5, D5.mm, It is noteorthy to mentioned that some dunes ere developed ith fine sediments (i.e. D5.58mm ) hich may not occur in natural flos,as reported by Balachandar et al. [5]. According to Ali and Lim [6], the minimum channel idth should not be less than ten times of gate opening. Considering the maximum gate opening as.5 centimeters, the selected idth of flume ould be acceptable since, 4/.5=6.4>. It is noteorthy to mention that some uncertainties may arise as the conseuences of secondary phenomena such as dune formation, unsteady flood flo, nonuniformity of sediment and time duration hich is the task of future researches. 4 LB 84.. Results and Discussion.. Maximum euilibrium depth of local scour Maximum euilibrium scour depth at donstream of adverse stilling basin due to the submerged jet issuing from a sluice gate, can be expressed in a functional form as: F,,, g,, LB, tan,, D5, s, g, Cu, tan, () here is the mass density of fluid, is the fluid kinematic viscosity, is the flo discharge per unit H. Khalili Shayan, J. Farhoudi, H. Hamidifar

5 [ DOI:.68/IJCE... ] idth, density of sediments, deviation of sediments, g s is the mass xm t 4hr is the geometric standard xm t 54hr is the repose angle of material, Cu is the sediments uniformly coefficient. Since the present study considers only uniformly graded sediments, E. () could be simplified as: F,,, g,, LB, tan,, D5, s, tan, () Using Buckingham Theorem, and taking the maximum euilibrium scour depth ( ) as a dependent parameter Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 the folloing euation is obtained: y L tan t B Fr 4 D5 5 6 Re () Since the out coming flo from the stilling basin as turbulent, therefore Re could be relaxed. In E. (), and i ; i 6 have to be determined experimentally in euilibrium conditions. Figure 4 shos the typical time dependent maximum scour depth ( ym ), the distance of maximum scour depth location from the end of rigid apron ( xm ), and the length of scour hole ( xs ) along ith the observation after 54 hours. It ould be concluded that xs t 4hr xs t 54hr ym t 4hr.86, ym t 54hr.875 and.846. These figures dictate that the scour hole reaches to an uasi-euilibrium stage after 4 hours leaving the scour process to continue ith removal of fine grains out of the hole. In this uasi-euilibrium stage, the maximum depth of scour hole becomes approximately constant, hile the scour profile continues to change. On the other hand, the time reuired to reach the euilibrium stage, depends on the flo characteristics (e.g. approaching Froude number and tailater depth), flo geometry and sediment size. Therefore, occurrence a complete euilibrium phase after 4 hours may not be a realistic assumption. Figure 4-b shos the rate of the extension of the maximum scour depth. It is seen that the scour process is very rapid at the beginning hile ithin the first five hours, the rate of maximum scour depth considerably decreases. So, it seems that 4 hours is plausible considering the laboratory limitations(hat do you ant to say?!!). Balachandar et al. [5] reported that the euilibrium scour profiles ere not attained even after 96 hours. Considering the above findings, the folloing relation as obtained from experimental data after 4 hours to estimate the maximum uasi-euilibrium scour depth as: y.8 tan. t.964 LB.695 Fr.465 D5.48 (4) Fig. 4 (a), (c), (d) Time variation of scour hole dimensions for 54 hr test, (b) Time rate of maximum scour depth The regression coefficient (R) beteen the experimental observations and the computed values as found to be.9 (standard error (SE)=.5), indicating that E. (4) satisfactorily agrees ith the experimental data. It should be noted that the E.4 is valid for the range of.9<tanθ<.56,.74</<.5, 4.<Fr<.4, 47.4<LB/<84.,.<D5/<.7. E. (4) demonstrates that, the maximum depth of scour increases International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5 5

6 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 ith incoming Froude number hile, decreases ith sediment size, length and slope of stilling basin. It can be also concluded that, the maximum scour depth donstream of adverse stilling basin decreases ith tailater depth hich contradicts ith the report Dey and Sarkar [6] for horizontal stilling basin (see Table ). From the experimental records donstream of horizontal rigid apron (zero slope), the folloing expression as revealed: y.7 t.98 LB.5 Fr.48 D5 (5).. Local scour donstream of horizontal stilling basins under submerged flo Figure 7-a shos the scour hole in the euilibrium state due to the submerged jet issuing from the sluice gate upstream a horizontal bed protection. The forces on a control volume, beteen the gate and donstream section, are as follo (see Fig. 7-b): The force due to the hydrostatic pressure distribution y, The force due to the hydrostatic pressure distribution at adjacent to the gate FP b, Weight of ater in control volume W b C.V., in hich is the specific eight of ater, b is the idth the donstream section FP b C.V. is the volume of fluid, u f is the flo velocity at the center of particle. It should be noted that the lift force depends on the same variables as the drag force and therefore the effect of the lift force is automatically taken into account. The drag force in the scour hole can be defined as (Graph [7]): FD CDbxse ub y and are the flo depths as shon in Fig. 7-a. The drag force acting on the bed of scour hole D uf, FD C D 4 in hich C D is the drag coefficient, D is the particle here xs is the length of the scour hole and ub is the flo velocity near the bed. It should be noted that the length of upstream face of the scour hole is relatively small compared ith the donstream side (Hoffmans []). It as observed that the upstream face of scour hole as attains / of the maximum length of scour hole after 48 hr as shon in Fig.. To define a relationship for length of scour hole, the x observed dimensionless length of scour hole ( se ) as depicted against the dimensionless maximum depth of y scour hole ( me ) at different slopes of adverse basin ( S o ) in Figs. 5 hich could be expressed in the form of xse y. me here The values of could be defined as xse. xse ith S o are plotted in Fig. 6 hich could be concluded as:.7a tan 44.6So 6.95 It is clear from E. (7) that the values of (7) xse ould be related to as: xse (8) The flo velocity near the bed could be related to the maximum flo velocity of submerged jet in a distance x from the gate ( um (x) ) as ub ku m (x). Hassan and Narayanan [] presented the folloing euations for estimation of the maximum flo velocity in the scour hole:.5.55 x um ( x) e.8.8u LB x, 4.8 y A A L, A.5.65 B.5. L x x, 4.8 um ( x)..8u B A A 6 (6).45 Euation (5) can be used for estimating the maximum scour depth donstream of horizontal stilling basin ith (R =.88) and (SE =.45) in the range of 5</<5.9,.78<Fr<.5, 47.4<LB/<84.,.<D5/<.7. It can be seen that the maximum scour depth donstream of horizontal bed protection increases ith the tailater depth hich is in agreement ith the euation previously reported by Dey and Sarkar [6]. of gate, size, (9) H. Khalili Shayan, J. Farhoudi, H. Hamidifar

7 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 Fig. 5 Determination of at different slopes of stilling basin Fig. 6 Variation of ith So International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5 7

8 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 Fig. 7 (a) Scour hole donstream of submerged sluice gate ith horizontal stilling basins, (b) Effective forces on the control volume after the gate here u The force due to the bed shear stress ( F ): Ead and Rajaratnam [8] reported a direct measurement of shear stress distribution due to the submerged jet issuing from a gate opening in shallo tailater conditions. They presented the folloing euation to estimate the shear force at any distance ( x ) from the gate ( F x ): is the flo velocity at the vena Cc contracta ( Cc Conseuently, ub ku FD is the contraction coefficient). CDb k u 4 () x x x x M F x here F ( x LB ) M bv b. Integrating E. () () gives the total bed shear force along the stilling basin as: 4 9 x 6 x 4 x x LB dx b I b x x. () here I is given as: 5 4 L L L L I. 9 B.68 6 B.8 4 B.6 B () The momentum euation beteen the sections adjacent to the gate and that of tailater could be expressed as: F x y Q Vx FP FP FD F Q V V b Introducing the foregoing relationships in E. (4) and assuming FP FP, a semi-theoretical euation could be obtained to estimate the maximum euilibrium depth of scour as: 8 (4) KCc Cc I Cc (5) H. Khalili Shayan, J. Farhoudi, H. Hamidifar

9 Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 [ DOI:.68/IJCE... ] here K is K CD k. It is noteorthy to mention that the flo discharge and the characteristics of the sediment are not included in E. (5). Using the Buckingham Theorem, the unknon K could be defined as K f,, D5, Ss f Fo. Using the experimental results, the K values are plotted against F in Fig. 8 hich could be expressed as: K.4F.779 (6) Finally a comparison is made beteen E. (5) and other researches related to horizontal aprons and results are presented in Figure 9 together ith Table. The ratio of the calculated maximum local scour depth to its y me cal experimental value ( r ), as used to evaluate y me exp the strength of each method in predicting the scour depths. Table shos that the euations proposed by Altinbilek and Basmaki [9] and Hoffmans [] may overestimate the maximum local scour depths. Whereas, the proposed euations by Hamidifar et al. [] and Chatterjee et al. [4], relating to a sufficient length of hydraulic jump, ould underestimate the maximum local scour depth in certain cases. It can be observed from Fig. 9 and Table that Es. (5) and(5) ould precisely predict the values of scour depth in hich the length of stilling basin is one of the effective parameter. The regression relationships and their parameters are extremely dependent on the range of experimental data and sensitive to any changes in the experiment conditions. While the momentum approach ould give a more general form of the phenomenon. Fig. 8 Variation of K ith densimetric Froude number Fig. 9 Comparison of the maximum euilibrium scour depth from the proposed relations against the experimental data from this research for (a) L LB 47.,59,84., (b) B 84. International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5 9

10 Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 [ DOI:.68/IJCE... ] Table Comparison beteen proposed euations based on experimental data from this study Proposed method E. (5) E. (5) Dey and Sarkar [6] Hamidifar et al. [] Chatterjee et al. [4] Altinbilek and Basmaki [9] Hoffmans [] r> Local scour donstream of adverse stilling basins under submerged flo.9<r< <r< r< stilling basin is shon in Figure -a. Neglecting the shear force on the adverse stilling basin, the forces acting on the control volume are (see Fig. -b): The euilibrium scour hole donstream of the adverse Fig. (a) Scour hole donstream of submerged sluice gate folloed by an adverse slope stilling basins, (b) Effective forces on the control volume donstream of the sluice gate FP FP The pressure force immediate to the gate; b ycos, The pressure force at the tailater section; b, The horizontal force along the stilling basin; Fx Wb Sin.Cos blb y Sin.Cos (7) is the bed angle, Wb is the eight of ater over the adverse stilling basin, and is the ratio of ater eight in the control volume to that is determined by assuming a linear profile for hydraulic jump. Pagliara and Peruginelli [], from their experimental ork, concluded that the values are neither dependent on bed slope nor on approaching Froude number and attains a constant value of.6. Assuming, The momentum euation in the horizontal direction may be ritten as: F x The drag force acting in scour hole; FD here c D k C D b k u FD b u y (8) is the flo depth immediately after the gate, FPx Fp Fx FP FD b V Vx (9)..Cos, V Introducing foregoing relationships in E. (9), results in a theoretical euation to estimate the maximum here Vx V.Cos H. Khalili Shayan, J. Farhoudi, H. Hamidifar

11 [ DOI:.68/IJCE... ] euilibrium local scour depth ( relative submergence ( s s as a function of y ), relative tailater depth ( angle of bed slope ( ), and approaching Froude number ( Fr ) as: g Cos ), dimensionless length of stilling basin ( LB /), LB Sin Cos s s s Cos Fr Cos s Fr Cos s Applying the energy euation before and after the gate and neglecting the energy losses, the relative submergence could be Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 ) s obtained as y me Fr Cos Fr Cos, here Cc ar Fr Cos Cc ar For the hydraulic jump on the horizontal stilling basin ( Fr Fr s Cc ar s ( ) hich can be compared ith E. (5). It is noteorthy to mention that, in Es. (),() and () the physical characteristics of sediment is not included. Using Buckingham Theorem, the unknon parameter could be expressed as f D5. Using the observed experimental data, the function of f D5 could be determined by the folloing euation (see Fig. -a):. As a result: yo Cos Fr Cos s LB Sin Cos Fr Cos s C a s c r ), E. () reduces to: y me Fr ar () f D5.8.5D5.8 D5 () Figure -b compares the accuracy of Es. (4),(5),(5)and() ith experimental data. Although the above semi-theoretical euation and the proposed regression relation (E. 4) have a similar predictability, but one could realize that the semi-theoretical euation has a more general form..4. Effects of different parameters on the maximum euilibrium scour depth The proposed semi-theoretical euation can be used to understand the effects of the geometry of stilling basin, the approaching Froude number and the tailater depth on maximum depth of scour. E. () shos that the () LB Sin. At the end of the adverse stilling basins, the streamlines incline toard the ater surface and then impinge toard the bed. As the length or bed slope of stilling basin increases, the excess shear stress decreases and the position of impinging point moves donstream of scour hole Which results in a reduced maximum scour depth donstream of the adverse stilling basins,as shon in Fig.. Figures. -b and -a sho the effect of tailater depth on the maximum scour depth for So.96 and So., respectively. For certain values of the densimetric Froude number ( Fo ), length, and slope of stilling basin, a g s s D5 maximum scour depth decreases ith critical value for tailater depth exists hich corresponds to a maximum value of euilibrium scour depth. It is shon that the scour depth increases ith the tailater depth reaching a maximum value here after decreases to attain a constant value. Figure 4 depicts the variations of the maximum scour depth as a function of the tailater depth donstream of the horizontal stilling basins for different apron lengths and densimetric Froude numbers. It is shon that a certain value of the tailater depth exists corresponding to a minimum value of the euilibrium scour depth hich as previously reported by Dey and Sarkar [6]. This may be attributed to instability of the hydraulic jump on the adverse slope stilling basins. McCoruodale and Mohamed [] reported that this type of jump is difficult to establish at Froude numbers loer than 9 and reuires continuous tailater adjustment to maintain a stationary position, at Froude numbers loer than about 4. The instability of the hydraulic jump over the adverse stilling basins could be balanced by the acting forces on the stilling basins. The hydraulic jump over the adverse slopes ould stand on stilling basin if the repelled forces (pressure force at the upstream side) are less than International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5

12 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 the counteracting forces (pressure force at the tailater section and the horizontal bed force along the stilling basin). At lo tailater depths, the hydrodynamic force, after the gate, ould be dominant. Conseuently, the hydraulic jump moves donstream and results in a considerable erosion. As the tailater depth increases, the pressure force at the end of the stilling basin increases. Therefore, the maximum scour depth decreases to reach a constant value. It is expected that this trend can be explained by the proposed semi-theoretical euation. Differentiating E. () ith respect to the relative tailater depth ( s tailater depth, d ym, one could achieve: ds L B Sin Cos D D y 6 (4) here D is: LB L Sin Cos 7 FrCos B Sin Cos y y, D p, p 9 54 Figure 5 depicts the effects of the stilling basin s length and bed slope on the critical tailater depth using E. (4) for Fr 9. It is shon that the critical tailater depth decreases ith the bed slope and increases ith the length of stilling basin. It should be mentioned that the ) and assuming that at critical (5) regression relationship (E. 4) cannot predict the to folds variations of scour depth ith tailater depth. Conseuently, the semi-theoretical euation is recommended to be used for prediction of maximum scour depth. Figure (a) f D5 as function of D5, (b) Comparison accuracy of proposed euations H. Khalili Shayan, J. Farhoudi, H. Hamidifar

13 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 Fig. Effects of slope and length of stilling basin on the maximum euilibrium local scour depth ( F, Fig. Effect of tailater depth on the maximum sour depth for (a) 6) So.96, (b) So. Fig. 4 Effect of tailater depth on the maximum sour depth for horizontal stilling basin International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5

14 [ DOI:.68/IJCE... ] Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 Fig. 5 Effects of slope and length of stilling basin on the critical tailater depth Fo 4. Conclusions FP Using the momentum principle, some theoretical euations ere derived to estimate the maximum depth of scour donstream of horizontal and adverse stilling basin. The proposed euations ere verified by experimental observations. The proposed semi-theoretical euations shoed an acceptable accuracy. The regression relationships and encountered parameters are extremely dependent on the range of experimental data and vary ith any changes in the experimental conditions. In the case of adverse stilling basins, it as shon that the scour depth increases ith tailater depth reaching a maximum value here after decreases to attain a constant value. Also, it as observed that the maximum depth of the scour hole decreases as the length and slope of stilling basin increases. The proposed semi-theoretical euation can be used to predict the critical tailater depth. In present ork, a semi-theoretical method based on momentum principle as presented to predict semi-maximum depth of scour hole donstream of horizontal and adverse stilling basins hich forms the core of the ork. Since the achievement has a theoretical basis, the proposed relationship could reflect the rate of influences induced by different parameters on the phenomenon. Therefore, the results ould be adapted to donstream of any hydraulic structures. Notation ar Relative gate opening ( ar b Gate idth Contraction coefficient Cc CD Cu D D5 FD 4 Drag coefficient Uniformly coefficient Sedimentological diameter 5% finer sediment size Drag force in the scour hole ) yo FP Fr F g LB So s Densimetric Froude number Force due to hydrostatic pressure distribution after the gate Force due to hydrostatic pressure distribution at the tailater section Approaching Froude number Force due to the bed shear stress Acceleration due to the gravity Length of stilling basin Flo discharge per unit idth of the rectangular basin Slope of stilling basin y Relative submergence of the sluice gate ( s ) ss Relative density of sediments s Relative tailater depth ( s uo Jet velocity at the gate opening Gate opening Weight of ater Weight of ater on the adverse stilling basin Horizontal distance of maximum scour depth from the stilling basin at time t Horizontal distance of maximum euilibrium scour depth from the stilling basin Length of scour hole at time t W Wb xm xme xs xse yo y ym ) Length of scour hole at euilibrium stage Flo depth at upstream of the gate Flo depth at the vena contracta Flo depth immediately after the gate Maximum scour depth at time t Maximum euilibrium local scour depth Tailater depth Critical tailater depth at the presence adverse stilling basin H. Khalili Shayan, J. Farhoudi, H. Hamidifar

15 Donloaded from ijce.iust.ac.ir at :9 IRST on Monday January 7th 9 [ DOI:.68/IJCE... ] s g Critical tailater depth at the presence horizontal stilling basin Angle of the bed slope Mass density of fluid Mass density of sediments Angle of repose Fluid kinematic viscosity Specific eight of ater Geometric standard deviation References [] [] [] [4] [5] [6] [7] Farhoudi J, Smith KVH. Time scale for scour donstream of hydraulic jump, Journal of Hydraulic Engineering, ASCE, 98, No., Vol. 8, pp Farhoudi J, Smith KVH. Local scour profiles donstream of hydraulic jump, Journal of Hydraulic Research, 985, No. 4, Vol., pp Hassan NMKN, Narayanan R. Local scour donstream of an apron, Journal of Hydraulic Engineering, ASCE, 985, No., Vol., pp Chatterjee SS, Ghosh SN, Chatterjee M. Local scour due to submerged horizontal jet, Journal of Hydraulic Engineering, ASCE, 994, No. 8, Vol., pp Balachandar R, Kells JA, Thiessen RJ. The effect of tailater depth on the dynamics of local scour, Canadian Journal of Civil Engineering,, Vol. 7, pp Dey S, Sarkar A. Scour donstream of an apron due to submerged horizontal jets, Journal of Hydraulic Engineering, ASCE, 6, No., Vol., pp Dey S, Sarkar A. Effect of upard seepage on scour and flo donstream of an apron due to submerged jets, ASCE, 7, No., Vol., pp [8] [9] [] [] [] [] [4] [5] [6] [7] [8] Oliveto G, Comuniello V, Bulbule T. Time-dependent local scour donstream of positive step stilling basins, Journal of Hydraulic Research,, No., Vol. 49, pp. 5-. Altinbilek HD, Basmaki Y. Localized scour at the donstream of outlet structures, Proceedings of the th congress of large dams, Madrid, 97, pp. 5-. Hoffmans GJCM. Jet scour in euilibrium phase, Journal of Hydraulic Engineering, ASCE, 998, No. 4, Vol. 4, pp Hamidifar H, Omid H, Nasrabadi M. Bed scour donstream of sluice gates, Journal of Water and Soils,, No. 4, Vol. 4, pp (in Persian). McCoruodale JA, Mohamed MS. Hydraulic jumps on adverse slopes, Journal of Hydraulic Research, 994, No., Vol., pp. 9-. Pagliara S, Peruginelli A. Limiting and sillcontrolled adverse-slope hydraulic jump, Journal of Hydraulic Engineering, ASCE,, No., Vol. 6, pp Rajaratnam N. The hydraulic jump in sloping channels, Irrigation and Poer, 966, No., Vol., pp Baines PG, Whitehead JA. On multiple states in single-layer flos, Physics of Fluids,, Vol. 5, pp. 98. Ali KHM, Lim SY. Local scour caused by submerged all jets, Proceedings of the Institute of Civil Engineers, 986, Part, Vol. 8, pp Graf WH. Hydraulics of Sediment Transport, McGraHill Book Co. Inc, Ne York, NY, 97. Ead SA, Rajaratnam N. Plane turbulent all jets in shallo tailater, Journal of Hydraulic Engineering, ASCE,, No., Vol. 8, pp International Journal of Civil Engineering, Vol., No., Transaction A: Civil Engineering, March 5 5